MAXIMIZATION AND MINIMIZATION PROBLEMS RELATED TO A p-laplacian EQUATION ON A MULTIPLY CONNECTED DOMAIN. N. Amiri and M.

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1 TAIWANESE JOURNAL OF MATHEMATICS Vol. 19, No. 1, pp , February 2015 DOI: /tjm This paper is available online at MAXIMIZATION AND MINIMIZATION PROBLEMS RELATED TO A p-laplacian EQUATION ON A MULTIPLY CONNECTED DOMAIN N. Amiri and M. Zivari-Rezapour* Abstract. In this paper we investigate maximization and minimization problems related to a p-laplacian equation on a multiply connected domain in R 2,where the admissible set is a rearrangement class of a fixed function. We prove existence and representation of the maximizers and existence, uniqueness and representation of the minimizer. 1. INTRODUCTION Let be a nonempty, bounded, connected open set in R 2 whose boundary is a disjoint union of simple closed curves Γ 0, Γ 1,...,Γ n of class C 2, and suppose Γ 0 encloses. Let1 <p<, we denote the conjugate of p by p = p p 1.Weconsider the following boundary value problem (1) Δ p u = f in, u =0 on Γ 0, u = constant on Γ i, i =1,...,n, Γ i u p 2 u n ds = γ i for i =1,...,n, where f L p (), n is the unit outer normal to, boundary of, andγ 1,...,γ n are real numbers. When p =2, G. R. Burton in [4, Appendix] has proved that the problem (1) has exactly one solution. By similar method, we show that (1) still has a unique solution when 1 <p<. For each f L p () we denote the unique solution of (1) by u f. For application of (1), when p = 2, to fluids dynamics (vorticity) see [4]. Our interest in this paper are in the maximization and minimization of the quantity 1 p u f p dx, the kinetic energy, as f varies in a rearrangement class of a fixed function in L p (); see the next section for precise definition of rearrangement of Received October 6, 2013, accepted June 4, Communicated by Franco Giannessi Mathematics Subject Classification: 35J20, 49J20. Key words and phrases: Rearrangement, Maximization, Minimization, Existence, Uniqueness. *Corresponding author. 243

2 244 N. Amiri and M. Zivari-Rezapour functions. Rearrangement optimization problems have been investigated in recent years by many authors, see [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. However, the boundary value problem in the one which is discussed here is interesting and very different from others. 2. REARRANGEMENT Let E be a (Lebesgue) measurable set in R N. Real measurable functions f and g on E are rearrangements of each other whenever L N ({x :f(x) α}) =L N ({x :g(x) α}), for all α R, where L N denotes the N-dimensional Lebesgue measure. It is well known that if f L r (E), 1 r,andg be a rearrangement of f, theng L r (E) and in fact f r = g r,where. r denotes the standard norm on L r (E). We denote the rearrangement class of f by R(f) which comprises all functions which are rearrangements of f. The readers can see [2, 3] for more results about rearrangements of functions. We now collect some useful lemmas to be applied later. Lemma 2.1. ([3]). Let p>1 and f 0 L p (E). Then (i) R(f 0 ), the weak closure of R(f 0 ) in L p (E), is compact with respect to L p - topology, weak topology, on L p (E). (ii) R(f 0 ) is convex. Lemma 2.2. ([3]). Let f 0 : E R and g : E R be two measurable functions. If every level set of g has measure zero then there exists an increasing function ξ such that ξ(g) R(f 0 ). Furthermore there exists a decreasing function η such that η(g) R(f 0 ). Lemma 2.3. ([3]). Let p>1, f 0 L p (E) and g L p (E). (i) If there is an increasing function ξ such that ξ(g) R(f 0 ) then fg dx ξ(g)g dx, for all f R(f 0 ), E and ξ(g) is the unique maximizer relative to R(f 0 ). (ii) If there is a decreasing function η such that η(g) R(f 0 ) then fg dx η(g)g dx, for all f R(f 0 ), E and η(g) is the unique minimizer relative to R(f 0 ). E E

3 Maximization and Minimization Problems Related to a p-laplacian Equation 245 Lemma 2.4. ([2]). Let 1 r and s be the conjugate exponent of r. Let g L r (E) and Ψ:L r (E) R be convex. (i) Suppose that Ψ is sequentially continuous in the L s -topology on L r (E). Then Ψ attains a maximum value relative to R(g). (ii) Suppose that Ψ is strictly convex, that g is a maximizer for Ψ relative to R(g) and that w is a member of sub-gradient of Ψ at g. Then g = ξ(w) almost everywhere in E for some increasing function ξ. 3. EXISTENCE AND UNIQUENESS In this section we prove the existence and uniqueness for the boundary value problem (1). Theorem 3.1. Let γ 1,...,γ n R, 1 <p<, and f L p (). boundary value problem (1) has a unique solution. Then the Proof. Let 0, 1,..., n be the regions enclosed by Γ 0, Γ 1,...,Γ n.let W = {w W 1,p () w =0on Γ 0 and w = constant on Γ i, i =1,...,n}. If w W, then we denote the value of w on Γ i by (w) i for i =1,...,n.Define J(w) := 1 p w p dx fw dx + γ i (w) i, w W. By the trace embedding W 1,p () L p ( ) we infer that W is a closed linear subspace of W 1,p (), andw comprises the restrictions to of elements of W 1,p 0 ( 0 ) that are constant on i, i =1,...,n. We consider the equivalent norm for W 1,p 0 ( 0 ) that is defined as follows ( ) 1 w = w p p dx. 0 It is well known that the function x x p, x R N, is strictly convex. From this, it is easy to deduce that J is strictly convex. We know that J is differentiable on W with J (w)v = w p 2 w v dx fv dx + γ i (v) i. Moreover, J(w) 1 p w p f p w p + γ i (w) i 1 p w p C w + γ i (w) i,

4 246 N. Amiri and M. Zivari-Rezapour for some C>0. Thus J is coercive because p>1. Therefore J has a unique global minimizer. We know that u W is a critical point of J whenever J (u)v =0,forall v W. Hence (2) u p 2 u v dx fv dx + γ i (v) i =0, for all v W. Let Lipschitz functions g 1,...,g n W be chosen to satisfy the boundary conditions (g j ) i = δ ij, i, j =1,...,n. Then (2) is equivalent to (3) (4) u p 2 u v dx u p 2 u g j dx fv dx =0, for all v W 1,p 0 (), fg j dx + γ j =0, j =1,...,n. Thus (3) is a variational formulation of Δ p u = f, in. Now from (4) and Divergence theorem we infer that ( Δ p u f)g j dx + g j u p 2 u n ds + γ j =0,j=1,...,n, Γ i which reduces to Γ j u p 2 u n ds + γ j =0, j =1,...,n. It follows that (1) holds if and only if u is a critical point of J, and therefore (1) has a unique solution. 4. OPTIMIZATION PROBLEM Let γ 1,...,γ n are fixed real numbers and 1 <p<. Also, let f 0 is a fixed function in L p () and R := R(f 0 ). It is well known that the solution u f of problem (1) satisfies the following variational problem (5) 1 u f p dx fu f dx + p γ i (u f ) i =min w W J f (w), where J f (w) := 1 w p dx p fw dx + γ i (w) i.

5 Maximization and Minimization Problems Related to a p-laplacian Equation 247 By (2), when v = u = u f, and (5) we deduce that (p 1) u f p dx =max ( pj f (w)) w W (6) ( =max p fw dx w W We define the functional ϕ : L p () R by ϕ(f):= u f p dx. Our interest is in the following optimization problems (7) max f R ϕ(f), and (8) min f R ϕ(f). We now prove some useful lemmas to be applied later. w p dx p ) γ i (w) i. Lemma 4.1. The functional ϕ is continuous with respect to weak topology in L p (). Proof. Let a sequence {f j } and f be all in L p () such that f j f in L p (). To simplify notation we write u j in place of u fj. From (6) we have (p 1)ϕ(f)+p (f j f)u f dx = p f j u f dx u f p dx p γ i (u f ) i (9) (p 1)ϕ(f j ) = p fu j dx u j p dx p γ i (u j ) i + p (f j f)u j dx (p 1)ϕ(f)+p (f j f)u j dx. Since f j f we deduce that (10) lim j (f j f)u f dx =0.

6 248 N. Amiri and M. Zivari-Rezapour Now, we prove that (11) lim j (f j f)u j dx =0. From (2), when v = u = u j, f j f, Poincarè s inequality for W 1,p 0 ( 0 ) and Hölder s inequality we find that u j p f j u j dx + γ i (u j ) i f j p u j p + C u j C u j, where C denotes a positive constant that can change from line to line. Note that in the above inequalities, the second inequality, we used this fact that if i {1, 2,,n} then (u j ) i p L 2 ( i )= u j p dx u j p dx C u j p. i 0 Hence {u j } W is a bounded sequence in W 1,p 0 ( 0 ), thus there exists a subsequence, still denoted {u j }, that converges weakly to û W, because W is closed. The compact imbedding of W 1,p () into L p () implies that {u j } converges strongly to û in L p (). Thus, we derive (11). Therefore, (9), (10) and (11) complete the proof of the lemma. Remark 4.1. According to the proof of the Lemma 4.1, we claim that û is equal to u f almost every where in. We know that (p 1)ϕ(f j )=p f j u j dx u j p dx p γ i (u j ) i. Now by the weak lower semicontinuity of the norm. and (6), we derive (p 1)ϕ(f) p fû dx û p dx p (p 1)ϕ(f). γ i (û) i Therefore, the uniqueness of the maximizer of the functional pj f (.) implies that û = u f almost every where in. Lemma 4.2. The functional ϕ is strictly convex in L p ().

7 Maximization and Minimization Problems Related to a p-laplacian Equation 249 Proof. Let 0 t 1 and f, g L p (). For each w W we have pj tf+(1 t)g (w)=p (tf +(1 t)g)w dx w p dx p Hence, from (6) we infer that = t( pj f (w)) + (1 t)( pj g (w)). ϕ(tf +(1 t)g) tϕ(f)+(1 t)ϕ(g); γ i (w) i thus ϕ is convex. Now, we show that ϕ is strictly convex. Suppose for some 0 <t<1, we have ϕ(h) =tϕ(f)+(1 t)ϕ(g), where h := tf +(1 t)g. Thus, Hence, J h (u h )=tj f (u f )+(1 t)j g (u g ). tj f (u h )+(1 t)j g (u h )=tj f (u f )+(1 t)j g (u g ). Since 0 <t<1, wederivej f (u h )=J f (u f ) and J g (u h )=J g (u g ). By the uniqueness of the minimizer of the functionals J f (.) and J g (.) on W, we deduce that u h = u f = u g, a.e. in. Thus, Δ p u f = Δ p u g almost every where in, sof = g almost every where in. Therefore, ϕ is strictly convex. Lemma 4.3. Let f L p (). The functional ϕ is Gâteaux differentiable at f with derivative ϕ (f)g = p gu f dx, p 1 for all g L p (). Proof. Let {t j } be a sequence of positive numbers that tends to zero. Let f, g L p () and h j := f + t j (g f), j 1. So,h j f in L p () as j 0. From (9) we have (p 1)ϕ(f)+p (h j f)u f dx (p 1)ϕ(h j ) (p 1)ϕ(f)+p (h j f)u j dx,

8 250 N. Amiri and M. Zivari-Rezapour where u j := u hj. Thus, p (g f)u f dx ϕ(f + t j(g f)) ϕ(f) p (12) (g f)u j dx. p 1 t j p 1 As a consequence of Remark 4.1, u j u f in L p (). This coupled with (12), implies that ϕ(f + t j (g f)) ϕ(f) lim = p (g f)u f dx. j t j p 1 Therefore, the proof of the lemma follows. Now, we are ready to prove the main results of this section. Theorem 4.4. The maximization problem (7) is solvable; that is, there exists f R such that ϕ(f )=max ϕ(f). f R Moreover, there exists an increasing function ξ such that f = ξ(u f ) almost everywhere in. Proof. From Lemma 4.1, Lemma 4.2 and Lemma 2.4(i) we infer that there exists f Rsuch that ϕ(f) ϕ(f ),forallf R. From Lemma 4.3, ϕ(f) is Gâteaux p differentiable with derivative p 1 u f. Since ϕ is strictly convex, by Lemma 2.4(ii), there is an increasing function ξ such that f = ξ(u f ). Theorem 4.5. If f 0 > 0 in, then the minimization problem (8) has a unique solution. Moreover, if f be the minimizer, then f = η(u f ) for some decreasing function η. Proof. We know ϕ is weakly continuous in L p (), Lemma 4.1, and R is weakly compact, Lemma 2.1. Thus, there exists f R such that ϕ(f )=minϕ(f). f R Since ϕ is strictly convex, Lemma 4.2, we infer that f is unique. Now, we prove that f R. From Lemma 2.14 of [3] we have L 2 ({x :f (x) > 0} L 2 ({x :f 0 (x) > 0} = L 2 (), so, f > 0 in. This coupled with Δ p u f = f in, implies that every level set of u f in has measure zero. By applying Lemma 2.2 we derive that there exists a decreasing function η such that η(u f ) R. Now, from Lemma 2.3(ii) we have (13) dx η(u f )u f dx, for all f R. fu f

9 Maximization and Minimization Problems Related to a p-laplacian Equation 251 Let 0 <t<1 and f R. Wedefinef t := tf +(1 t)f.sincerisconvex, Lemma 2.1(i), f t R for all 0 <t<1. From Lemma 4.3, for sufficiently small t we have ϕ(f ) ϕ(f t )=ϕ(f )+ tp (f f )u f dx + o(t). p 1 Thus, when t 0 + we deduce (14) dx fu f f u f dx, for all f R. Therefore, by (13), (14) and Lemma 2.3(ii) we derive f = η(u f ). REFERENCES 1. R. Adams, Sobolev Spaces, Academic Press, New York, G. R. Burton, Rearrangements of functions, maximization of convex functionals, and vortex rings, Math. Ann., 276 (1987), G. R. Burton, Variational problems on classes of rearrangements and multiple configurations for steady vortices, Ann. Inst. Henri Poincare, 6 (1989), G. R. Burton, Rearrangements of functions, saddle points and uncountable families of steady configurations for a vortex, Acta Math., 163 (1989), F. Cuccu, B. Emamizadeh and G. Porru, Optimization of the first eigenvalue in problems involving the p-laplacian, Proc. Amer. Math. Soc., 137(5) (2009), F. Cuccu, B. Emamizadeh and G. Porru, Optimization problems for an elastic plate, J. Math. Phys., 47(8) (2006), , 12 pp. 7. F. Cuccu, B. Emamizadeh and G. Porru, Nonlinear elastic membranes involving the p-laplacian operator, Electron. J. Differential Equations, 49 (2006), 10 pp. 8. F. Cuccu, G. Porru and S. Sakaguchi, Optimization problems on general classes of rearrangements, Nonlinear Analysis, 74 (2011), F. Cuccu, G. Porru and A. Vitolo, Optimization of the energy integral in two classes of rearrangements, Nonlinear Stud., 17(1) (2010), L. M. Del Pezzo and J. Fernández Bonder, Some optimization problems for p-laplacian type equations, Appl. Math. Optim., 59(3) (2009), L. M. Del Pezzo and J. Fernández Bonder, Remarks on an optimization problem for the p-laplacian, Appl. Math. Lett., 23(2) (2010), B. Emamizadeh and M. Zivari-Rezapour, Optimization of the principal eigenvalue of the pseudo p-laplacian operator with Robin boundary conditions, International Journal of Mathematics, 23(12) (2012), , 17 pp.

10 252 N. Amiri and M. Zivari-Rezapour 13. B. Emamizadeh and M. Zivari-Rezapour, Rearrangements and minimization of the principal eigenvalue of a nonlinear Steklov problem, Nonlinear Anal., 74(16) (2011), B. Emamizadeh and M. Zivari-Rezapour, Rearrangement optimization for some elliptic equations, J. Optim. Theory Appl., 135(3) (2007), B. Emamizadeh and J. V. Prajapat, Maximax and minimax rearrangement optimization problems, Optim. Lett., 5(4) (2011), M. Zivari-Rezapour, Maximax rearrangement optimization related to a homogeneous Dirichlet problem, Arab. J. Math., 2(4) (2013), , DOI /s N. Amiri and M. Zivari-Rezapour Department of Mathematics Faculty of Mathematical Sciences & Computer Shahid Chamran University Golestan Blvd. Ahvaz Iran n-amiri@phdstu.scu.ac.ir mzivari@scu.ac.ir